12.5 Use of Shock Response Spectra

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12.5.1 Severity Comparison of Several Shocks

A shock, A, is regarded as more severe than a shock, B, if it induces in each resonator a larger stress.

One then carries out an extrapolation, which is certainly open to criticism, by supposing that, if

shock A is more severe than shock B when it is applied to all the standard resonators, it is

also more severe with respect to an arbitrary real structure (which cannot be linear nor have a

single DoF).

12.5.2 Test Specification Development from Real Environment Data

12.5.2.1 Synthesis of Spectra

Let us consider the most complex case where the real environment, described by curves of acceleration

against time, is supposed to be composed of p different events (handling shock, inter-stage cutting shock

on a satellite launcher, etc.), with each one of these events itself characterized by ri successive

measurements.

These ri measurements allow a statistical description of each event. The following procedure holds for

each one (Lalanne, 2002b; see Figure 12.22).

* Calculate the SRS of each signal recorded with the damping ratio of the principal mode of the

structure if this value is known, if not, use the conventional value 0.05. In the same way, the

12-18 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

frequency band of analysis will have to envelop the principal resonant frequencies of the structure

(known or foreseeable frequencies).

* If the number of measurements is sufficient, calculate the mean spectrum, m (mean of the

points at each frequency) as well as the standard deviation spectrum (s), then the mean

spectrum þ a standard deviations, according to the frequency; if it is insufficient, make the

envelope of the spectra.

The value of a can be either arbitrary (for example 2.5 or 3) or the result of a statistical

calculation. It is often considered that the SRS amplitudes obey to a log-normal distribution. If yi is

the logarithm of the SRS amplitude, yj ¼ log10 SRSj; the real environment envelope (for a given

probability P0 of not exceeding at the confidence level p0) can be defined by

SRSEnv ¼ 10my þasy ð12:14Þ

where my and sy are, respectively, the mean and the standard deviation of the yj values:

my ¼

1

ri

Xri

j¼1

yj ð12:15Þ

sy ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Xri

j¼1 ðyj 2 my Þ2

ri 2 1

vuuuuut

ð12:16Þ

The number of standard deviations, a; is given in Table 12.4 for different values of ri; P0; and p0:

SRSEnv can also be defined as the upper one-sided normal tolerance interval for which 100 P0% of

the values will lie below the limit with 100p0% confidence.

* Apply the mean spectrum or the mean spectrum þ a standard deviations a statistical uncertainty

coefficient (Lalanne, 2002d), calculated for a probability of tolerated maximum failure (taking

into account the uncertainties related to the dispersion of the real environment and of the

mechanical strength), or contractual (if one uses the envelope).

Event # 1

Handling shock

r1 measured data

Event # 2

Loading shock

r2 measured data

Event # p

Ignition shock

rp measured data

Calculation of

r1 S.R.S.

Mean and standard

deviation spectra

or envelope

Mean and standard

deviation spectra

or envelope

Mean and standard

deviation spectra

or envelope

k (m + 3 s)

or k × env.

k (m + 3 s)

or k × env.

k (m + 3 s)

or k × env.

Calculation of

r2 S.R.S.

Calculation of

rp S.R.S.

Envelope

Envelope

×

Test factor

FIGURE 12.22 Process of developing a specification from real shocks measurements.

Mechanical Shock 12-19

© 2005 by Taylor & Francis Group, LLC

Each event thus being synthesized in only one spectrum, one proceeds to an envelope of all the spectra

obtained to deduce from it an SRS covering the totality of the shocks of the life profile. After

multiplication by a test factor, which takes account of the number of tests performed to demonstrate the

resistance of the equipment (Lalanne, 2002d), this spectrum will be used as reference “real environment”

for the determination of the specification.

The reference spectrum can consist of the positive and negative spectra or the envelope of their

absolute value (maximax spectrum). In this last case, the specification will have to be applied according

to the two corresponding half-axes of the test item.

12.5.2.2 Nature of the Specification

There is an infinity of shocks having a given response spectrum. This property is related to the very great

loss of information in computing the SRS, since one retains only the largest value of the response

according to the time to constitute the SRS at each natural frequency.

According to the characteristics of the spectrum and available means, the specification can be

expressed in the forms given below.

* It can be a simple shape signal according to the time realizable on the usual shock machines

(half-sine, TPS, rectangular pulse).

One can thus try to find a shock of simple form, in which the spectrum is closed to the reference

spectrum, characterized by its form, its amplitude, and its duration. It is in general desirable that

the positive and negative spectra of the specification, respectively, cover the positive and negative

spectra of the field environment. If this condition cannot be obtained by application of only one shock

(owing to the particular shape of the spectra, and the limitations of the facilities), the specification will be

made up of two shocks, one on each half-axis. The envelope must be approaching the reference SRS as

well as possible, if possible on all the spectrum in the frequency band retained for the analysis, if not in a

frequency band surrounding the resonant frequencies of the test item (if they are known).

* It can be a SRS. In this case, the specification is directly the reference SRS.

12.5.2.3 Choice of Shape

The choice of the shape of a shock is carried out by a comparison of the shapes of the positive and

negative spectra of the real environment with those of the spectra of the usual shocks of simple shape

(half-sine, TPS, rectangle; Figure 12.23).

TABLE 12.4 Number of Standard Deviations Corresponding to a Given Probability of not Exceeding, P0; at the

Confidence Level, p0

ri \P0 p0 ¼ 0:75 p0 ¼ 0:90 p0 ¼ 0:95

0.75 0.90 0.95 0.99 0.75 0.90 0.95 0.99 0.75 0.90 0.95 0.99

3 1.464 2.501 3.152 4.396 2.602 4.258 5.310 7.340 3.804 6.158 7.655 10.552

4 1.256 2.134 2.680 3.726 1.972 3.187 3.957 5.437 2.619 4.416 5.145 7.042

5 1.152 1.961 2.463 3.421 1.698 2.742 3.400 4.666 2.149 3.407 4.202 5.741

6 1.087 1.860 2.336 3.243 1.540 2.494 3.091 4.242 1.895 3.006 3.707 5.062

7 1.043 1.791 2.250 3.126 1.435 2.333 2.894 3.972 1.732 2.755 3.399 4.641

8 1.010 1.740 2.190 3.042 1.360 2.219 2.755 3.783 1.617 2.582 3.188 4.353

9 0.984 1.702 2.141 2.977 1.302 2.133 2.649 3.641 1.532 2.454 3.031 4.143

10 0.964 1.671 2.103 2.927 1.257 2.065 2.568 3.532 1.465 2.355 2.911 3.981

15 0.899 1.577 1.991 2.776 1.119 1.866 2.329 3.212 1.268 2.068 2.566 3.520

20 0.865 1.528 1.933 2.697 1.046 1.765 2.208 3.052 1.167 1.926 2.396 3.295

30 0.825 1.475 1.869 2.613 0.966 1.657 2.080 2.884 1.059 1.778 2.220 3.064

40 0.803 1.445 1.834 2.568 0.923 1.598 2.010 2.793 0.999 1.697 2.126 2.941

50 0.788 1.426 1.811 2.538 0.894 1.560 1.965 2.735 0.961 1.646 2.065 2.863

Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.

12-20 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

12.5.2.4 Amplitude

The amplitude of a shock is obtained by

plotting the horizontal straight line that closely

envelops the positive reference SRS at high

frequency (Figure 12.24). This line cuts the

y-axis at a point which gives the amplitude

sought (here, one uses the property of the

spectrum at high frequencies, which tends in

this zone towards the amplitude of the signal in

the time domain).

12.5.2.5 Duration

The shock duration is given by the coincidence

of a particular point of the reference spectrum

(Figure 12.24) and the reduced spectrum of the

simple shock selected above.

One in general considers the abscissa, f01, of

the first point which reaches the value of

the asymptote at the high frequencies (amplitude

of shock) as shown in Figure 12.25. Table 12.5

joins together some values of this abscissa for the

most usual simple shocks according to the Q

factor (Lalanne, 2002b). Another possibility is to

use the coordinates of the first (higher) peak of the SRS, as given in Table 12.6.

Notes:

1. If the calculated duration must be rounded (in milliseconds), the higher value should always be

considered, so that the spectrum of the specified shock remains always higher or equal to the

reference spectrum.

Envelope at the high frequencies

Real environment

x = 0.05

200 300 400

Frequency (Hz)

100

49.5

0

−400

−300

−200

−100

0

S.R.S. (m/s2)

100

200

300

340

400

500

FIGURE 12.24 Determination of the amplitude and duration of the specification. (Source: Lalanne, Chocs

Mecaniques, Hermes Science Publications. With permission.)

S.R.S.

S.R.S.

S.R.S.

Rectangle

Q = 10

T.P.S.

Half- sine

f

f

f

FIGURE 12.23 Shapes of the SRS of the realizable

shocks on the usual machines. (Source: Lalanne, Chocs

Mecaniques, Hermes Science Publications. With permission.)

Mechanical Shock 12-21

© 2005 by Taylor & Francis Group, LLC

2. It is in general difficult to carry out shocks of

duration lower than 2 msec on standard

shock machines (except for very light

equipment).

One will validate the specification by checking

that the positive and negative spectra of the

shock thus determined will envelop the respective

reference spectra and one will verify, if the

resonant frequencies of the test item are known,

that one does not overtest exaggeratedly at these

frequencies.

Example

As an example, let us consider the positive and

negative spectra characterizing the real environment

plotted in Figure 12.24, which is a result of a

true synthesis. It is noted that the negative

spectrum preserves a significant level throughout

the entire frequency domain (the beginning of the

spectrum being excluded).

The most suitable simple shock shape is the TPS. The shock amplitude, whatever its waveform,

is equal to 340 m/sec2. The abscissa, f01, of the first point that reaches the value of the

asymptote is equal to 0.415. From this value, f0 ¼ 49:5 Hz; the duration is given by t ¼

0:415=49:5 ¼ 0:0084 sec:

The duration of the shock will thus be (rounding up) t ¼ 9 msec; which slightly moves the

spectrum towards the left and makes it possible to better cover the low frequencies. Figure 12.26

shows the spectra of the environment and those of the TPS pulse thus determined.

The main steps of deriving a shock test specification from the SRS of a real environment are outlined

in Table 12.7.

T.P.S. dimensionless S.R.S.

1.5

1.0

0.5

0.0

−0.5

−1.0

−1.5

0.0

0.415

x= 0.05

1.0 2.0 3.0 4.0

S.R.S. / xm

f0t

:

FIGURE 12.25 Determination of the shock duration.

(Source: Lalanne, Chocs Mecaniques, Hermes Science

Publications. With permission.)

TABLE 12.5 Values of the Dimensionless Frequency Corresponding to the First

Passage of the SRS by the Amplitude Unit

Q j f01

Half-sine TPS Rectangle

3 0.1667 0.358 0.564 0.219

4 0.1250 0.333 0.499 0.205

5 0.1000 0.319 0.468 0.197

6 0.0833 0.310 0.449 0.192

7 0.0714 0.304 0.437 0.188

8 0.0625 0.293 0.427 0.185

9 0.0556 0.295 0.421 0.183

10p 0.0500 0.293 0.415 0.181

15 0.0333 0.284 0.400 0.176

20 0.0250 0.280 0.392 0.174

30 0.0167 0.276 0.385 0.172

40 0.0125 0.274 0.382 0.170

50 0.0100 0.272 0.379 0.170

1 0.0000 0.267 0.371 0.167

(p) Conventional value

12-22 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

12.5.2.6 Difficulties

The response spectra of shocks measured in the real environment often have a complicated shape

which is impossible to envelop by the spectrum of a shock of simple shape realizable with the usual

test facilities of the drop table type. This problem arises in particular when the spectrum presents an

important peak (Smallwood and Witte, 1972). The spectrum of a shock of simple shape will be (see

Figure 12.27): either an envelope of the peak, which will lead to significant overtesting compared

with the other frequencies, or envelope of the spectrum except the peak, consequently leading to

undertesting at the frequencies close to the peak, if one knows that the material does not have any

resonance in the frequency band around the peak. The simulation of shocks of pyrotechnic origin

leads to this kind of situation.

TABLE 12.6 Values of the Dimensionless Frequency and Amplitude Corresponding to the First Peak of

the SRS

Q j Half-sine TPS Rectangle

f0 peak SRS Peak f0 peak SRS Peak f0 peak SRS Peak

3 0.1667 0.875 1.4249 0.723 1.0462 0.508 1.5880

4 0.1250 0.86 1.4958 0.700 1.0894 0.504 1.6731

5 0.1000 0.845 1.5425 0.688 1.1182 0.503 1.7292

6 0.0833 0.840 1.5757 0.681 1.1387 0.502 1.7690

7 0.0714 0.830 1.600 0.676 1.1541 0.502 1.7985

8 0.0625 0.83 1.6194 0.672 1.1660 0.501 1.8214

9 0.0556 0.830 1.6346 0.669 1.1755 0.501 1.8396

10p 0.0500 0.826 1.6470 0.667 1.1832 0.501 1.8545

15 0.0333 0.820 1.6854 0.661 1.2073 0.501 1.9005

20 0.0250 0.820 1.7054 0.658 1.2199 0.501 1.9244

30 0.0167 0.815 1.7258 0.656 1.2328 0.501 1.9490

40 0.0125 0.810 1.7363 0.654 1.2393 0.501 1.9615

50 0.0100 0.813 1.7426 0.653 1.2433 0.501 1.9691

1 0.0000 0.810 1.7685 0.650 1.2596 0.500 2.000

(p) Conventional value

T.P.S.

Real environment

x = 0.05

Frequency (Hz)

500

400

300

200

100

−100

−200

−300

−400

0

0 100 200 300 400

S.R.S. (m/s2)

FIGURE 12.26 SRS of the specification and of the real environment. (Source: Lalanne, Chocs Mecaniques, Hermes

Science Publications. With permission.)

Mechanical Shock 12-23

© 2005 by Taylor & Francis Group, LLC

Shock pulses of simple shape (half-sine, TPS) have, in logarithmic scales, a slope of 6 dB/octave

(i.e., 458) at low frequencies incompatible with those larger ones, of spectra of pyrotechnic shocks

(. 9 dB/octave). When the levels of acceleration do not exceed the possibilities of the shakers, simulation

with control using spectra is of interest (Section 12.10).

Note: In general, it is not advisable to choose a simple shock shape as a specification when the real

shock is oscillatory in nature. In addition to overtesting at low frequencies (the oscillatory shock is with

very small velocity change), the amplitude of the simple shock thus calculated is more sensitive to the

value of the Q factor in the intermediate frequency range.

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